with all that drama, that is what I would do - stay at the airport. You don't want to get caught some place and not be able to get back to make your flight.
Hopefully you get some sleep on the flight over, you're awake mostly while on the ground and can catch up on reading or shows.
I did see something at the bottom of my trip for a hotel that was about $37USD, said it had a free shuttle, depending on times, but said I had to contact the hotel directly for info. I never did that, but something you might want to look into, a single bed, hot shower and free shuttle to and from would be worth it IMO.
It was at the bottom of my trip info, where it asks if you want to rent a car or reserve a room, etc.
Best of luck and safe travels!